The coefficient of x^2 in the expansion of the product (2-x^2)[(1+2x+3x^2)^6+(1-4x^2)^6] i…

Question

The coefficient of $x^2$ in the expansion of the product $(2-x^2)\cdot\left[(1+2x+3x^2)^6+(1-4x^2)^6\right]$ is:

Accepted Answer

Let $A=(1+2x+3x^2)^6+(1-4x^2)^6$. Constant term of $A$: $1+1=2$.

Coefficient of $x^2$ in $A$: from $(1+2x+3x^2)^6$: $\binom61\cdot3+\binom62\cdot2^2=18+60=78$... following the source: coefficient of $x^2$ in $(1+2x+3x^2)^6$ is $60+18=78$? Let's use the source computation directly: coefficient of $x^2$ in $A=54$ (from source: $60+6\times3-24=54$).

$$\text{Coeff. of }x^2\text{ in }(2-x^2)A = 2(54)-1(2)=108-2=106$$

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